Question

Solve the matrix equation X-A=B for X if a= [1 -4 -7 4 7 6] and b= [-9 4 8 2 5 2].
X=

Answer 1

`\color{red} {{{\Large {\bf{To\:\:Simplify\::(\sin ^4(x)-\cos ^4(x))/(\sin ^2(x)-\cos ^2(x))}}}}}`

`\color{green}{{{\large {\bf{Your\:\:Answer\::(\sin ^4(x)-\cos ^4(x))/(\sin ^2(x)-\cos ^2(x))=1}}}}}`

`\color{yellow} {\Huge {\sf{Solution:}}}`

`\color{blue} {\large {\bf{Factor\:\sin ^4(x)-\cos ^4(x)}}}`

`\tt \color{blue} {\mathrm{Rewrite\:}\sin ^4(x)-\cos ^4(x)\mathrm{\:as\:}(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x))^2-(\cos ^2(x))^2}`

`\color{fuchsia} {\\ormalsize {\mathrm{Apply\:exponent\:rule}:\quad \:a^(bc)=(a^b)^c}}`

`\color{fuchsia} {\\ormalsize \sin ^4(x)=(\sin ^2(x))^2}`

`\color{fuchsia} {\\ormalsize =(\sin ^2(x))^2-\cos ^4(x)}`=

`\color{fuchsia} {\\ormalsize \mathrm{Apply\:exponent\:rule}:\quad \:a^(bc)=(a^b)^c}`

`\color{fuchsia} {\\ormalsize \cos ^4(x)=(\cos ^2(x))^2}`

`\color{fuchsia} {\\ormalsize =(\sin ^2(x))^2-(\cos ^2(x))^2}`=

`\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}`x^2-y^2=(x+y)(x-y)

`(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))`

=
`(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))`=

`\color{blue} {\large {\bf{Factor\:\sin ^2(x)-\cos ^2(x)}}}`

`\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}`x^2-y^2=(x+y)(x-y)

`\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)-\cos (x))`

(x)=(sin(x)+cos(x))(sin(x)−cos(x))

=(\sin (x)+\cos (x))(\sin (x)-\cos (x))=(sin(x)+cos(x))(sin(x)−cos(x))

`\large=(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))</p><p> (x))(sin(x)+cos(x))(sin(x)−cos(x))</p><p>\large =((\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x)))/(\sin ^2(x)-\cos ^2(x))`=

`\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}`x^2-y^2=(x+y)(x-y)

`\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)`

`(x)=(sin(x)+cos(x))(sin(x)−cos(x))</p><p>=((\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x)))/((\sin (x)+\cos (x))(\sin (x)-\cos (x)))`=

`\mathrm{Cancel\:}((\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x)))/((\sin (x)+\cos (x))(\sin (x)-\cos (x))):\quad \sin ^2(x)+\cos ^2(x)Cancel </p><p>(sin(x)+cos(x))(sin(x)−cos(x))`

`\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)+\cos(x)Cancelthecommonfactor:sin(x)+cos(x)`

=
`((\sin ^2(x)+\cos ^2(x))(\sin (x)-\cos (x)))/(\sin (x)-\cos (x))`=

`\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)-\cos`

`\mathrm{Use\:the\:following\:identity}:\quad \cos ^2(x)+\sin`

`\huge \boxed{\color{red} {\ \huge =1}}`